SS수학고등지도서(3-3)-13-OK

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1 3 g.06 g > 3.06>3 < 3+.06< { }< <{ } 3 3<.06< æ90 500_3+00_ < a<b a- b- -3a+ -3b+ 5-a 5-b ;5A; ;5B; < > > < -3æ5+6 -(-)<3- -7 ;!;-3 6-5> >: 5 : æ >7 3-5æ 3-5æ- [ [ >- +7<5 5->3 [ 3-< (-) - æ < 5 - < 8

2 a>b a-b>0.06>3 < < >.05.05< <.353. g a>b HjK a-b>0 a=b HjK a-b=0 a<b HjK a-b<0 a b a>b HjK a >b a b aæb HjK ;ba;æ a>b -a -b a>b -a a+(-a)>b+(-a)0>b-a -b -b+0>-b+(b-a) -b>-a -a<-b a>b-a<-b p. a>b>0 HjK ;b!;>;a!;>0a<b<0 HjK ;b!;<;a!;<0 a>b a+0 a a>b a=0 bæ0 b<0 85

3 b>c jk a-c>a-b b>c -b<-c a a-b<a-c yy a-c>a-b (a-c)-(a-b)=-c+b =b-c>0 b>c a-c>a-b a>b>0c>d>0 jk ;da;>;cb; a>b>0c>0 ac>bc>0 yy c>d>0b>0 bc>bd>0 yy ac>bc>bd>0 c>0d>0 cd>0 yy ac bc bd > > >0 yy cd cd cd c>0 ;da;>;db;>;cb;>0 y ;da;>;cb; ac-bd ;da;-;cb;= cd ac>bd c>0d>0 ac-bd>0 cd>0 yy ac-bd>0cd>0 ac-bd >0 yy cd ac-bd cd =-;da;-;cb;>0 a>b>0 jk a >b a<0b<0 -<0 -a>0-b>0 yy a>b -<0 -b>-a>0 yy (-a) <(-b) 0>a>b jk a <b -a + -b >c 0 ab a<b <a -(-a)-(-b)>c a <b +(-a)-(-b)>c æb +(-a)+(-b)>c 86

4 ;3$; -3 >7-3 >;&; -3<-;&; -3>;&; <-;!; >: : -a <0-0<-a<0-5+;A;<<5+;A; b<<6 5+;A;=6, b=-5+;a; a=b=- 3 0 < <5æ5 < =-(-)-(-5) 3 =-+7æ ;#; ;#; <;!; - <3+ - -(-)<3-(-) >-3-3<<;!; -3 < =(-)-(-5)=3 3< æ =-+-5 =-7æ 5 ;!; < - <3+ - -<3-(-) <;3%; ;!; < 5 3 æ: : æ: : æ ;#; æ: : <;!;;!; <æ - <3+ - -<3+(-) <3 <3 3-3<<3 87

5 pqr p q r a= p-q + q-r = p-r b= p-q + p-r c= p-r + q-r a a b c 0<a< a>0 -a>0 a(-a)>0a-a >0 a <a - +5æ9 <;!; -(-)+5æ9 -;#; æ;!; (-)+5æ9 æ;%; -;#; æ;%; -;#; æ-;%; +> >- <00 <æ <0-+(-)>- ->- 0 <+(-)>- >;!; ;!;<< 3-5 æ-(-)>- >- -3 < æ >;!; ;!;<< 3- æ ;!; æ;%; <-æ-5-5 <- - <0 6 - <0 æ0 0 < -5 88

6 - - a +b+c>0 a +b+c=0-5+>0 y= -5+ y>0 y= -5+ y= -5+ < > -5+<0 m -0. +(m) 8m =.6(m) y= -5+ y<0 y= -5+ y= -5+ <<. m 89

7 >0(-)(-7)> (-)(-7) < (-)(-7) + = 0-0 << = > >0 < > -- 0(-)(+) (-)(+) <- =- -<< = > + - (-)(+) (-)(+) 0 - a +b+c=0 D=b -ac D>0D=0D<0 a +b+c=0(a>0)ab(a<b) a +b+c=a(-a)(-b)=0 a +b+c>0 a +b+c<0 <a >b a<<b Oughtred, w >< Harriot, T

8 D>0-3-0<0 (-5)(+)<0 -<< (-)(-)æ0 ;!; æ D=0-8+6>0 (-) > (-) 0 = - +-9>0 (-3) < (5-) æ0 ~ _ 63 >< æ Bouguer, P E. 9

9 D=0 -k+ 0=a D=b -ac =(-k) - =6k -6 6k -6=0k = k= k= (-) =0 = k=- (+) =0 =- k= [ [ k=- a= a=- D<0-3+0=0 D=(-3) - 0=-3<0 >0 - +-=0 D=(-) - (-) (-)=-<0 <0 3-3+=0 D=(-3) - 3 =-5<0 3>0 -+3=0 D=(-) - 3=-8<0 >0 6 3 a +b+c>0 a>0d=b -ac<0 -a+a=0 D=(-a) - a =a -a<0 a(a-)<0 0<a< [ -5+<0 æ -5+=(-)(-)<0 << < -3-0<0 [ (+)(-5)<0 -<<5 (-)(-)æ0 ;!; æ -< ;!; <5-5 9

10 7 D æ0 -k+=0 D =k -=(k-)(k+)æ0 k - kæ -k+3k=0 D =k -3k=k(k-3)æ0 k 0 kæ3 k - kæ3-3-8=(-6)(+3) =(-6) 0 =6 -+3æ [ -5-6<0-5-6=(-6)(+)<0 -<<6 -< -5-0 [ -3 +-<0-5-=(-7)(+) =(3-)(-)>0 <;3!; > - <;3!; < 7 3 D D D =-(a+)<0 D =a +a-5<0 a>0-5<a< 0<a< +5æ3 ->0 [ [ +-< < - <0 +a+b<0 -<<3 a --b<0 -<<3 (+)(-3)<0 --6= +a+b a=-b= < =-( +-6) =-(+3)(-)<0 (+3)(-)>0 <-3 > 93

11 a a æ0 a =a a æ0 ææ a>0b>0 a+b ab æ'aåbæ a+b a=b A a B b C c a+b=;!; t +;!;(-t) ={t -t+;!;} c=(-t)t=(t-t ) t A t C -t (-t) B a+b-c={t-;!;}æ0 a+bæc t=;!; (-) æ0 - +>0 ab a>b HjK a-b>0 a æ0 a=0 a +b æ0 a=b=0 a =a ab = a b a>0b>0a>b HjK a >b ab a>b HjK a«>b«ab a>b HjK a«>b«n ab a>b HjK «'a>«'b ab a>b HjK «'a>«'bn ab a>b HjK ;ba;>>;ab; 9

12 a æ0 a -ab+3b =(a -ab+b )+b =(a-b) +b (a-b) `æ0b æ0 a -ab+3b æ0 a=b=0 a +ab+5b =(a +ab+b )+b =(a+b) +b (a+b) æ0b æ0 a +ab+5b æ0 a=b=0 9 a æ0a>b HjK a-b>0 a +-a=(a-) æ0 a +æa a= (a+b) a +b - =;!;{a +b -(a +ab+b )}=;!;(a-b) æ0 a +b æ (a+b) a=b 9 a>0 a+;a!;ææ a ;a!;= a+;a!;æ a=;a!; a= {a+;a!;}{a+;a!;}=a +++ {a+;a!;}{a+;a!;}=a + +5 a a>0a >0 >0 a a + ææ a = a a a + +5æ+5=9 a {a+;a!;}{a+;a!;}æ9 a ' a = a= a 9 5 a æ0a>b HjK a-b>0 ab 'aåb- = {(a+b)'aåb-('aåb) } a+b a+b = {'aåb(a+b-'aåb)} a+b = {'aåb('a-'b) } a+b a>0b>0 a+b>0'aåb>0 ('a-'b) æ0 {'aåb('a-'b) }>0 a+b 'aåbæ ab a+b 'a='b a=b 95

13 50 a æ0aæ0bæ0 a>b HjK a >b ( a-b ) -( a + b ) =(a-b) -(a + a b +b ) =-ab- ab ab æab -ab- ab 0 a-b a + b -ab= ab ab 0 "ça +"çb = a + b ( a + b ) -(a +b ) =a + a b +b -a -b = ab æ0 " a +" b æ" a +b ab=0 a +b -ab=(a-b) æ0 a +b æab a=b a -3ab+3b =[a - a ;#;b+{;#;b}]+;#;b a -3ab+3b ={a-;#;b}+;#;b {a-;#;b}æ0;#;b æ0 {a-;#;b}+;#;b æ0 a -3ab+3b `æ0 a=b=0 a+0a >0 >0 a a + ææ a = a a a = a= a a>0b>0 a+b æ'aåb a+b ab = æ'aåb a+b=ab a b ab HjjK æab HjjK æ HjjK abæ a=b= a +b æab a +b (a-b) -ab= æ0 a=b >0{+;[!;}(+) >0 {+;[!;}(+)=++;[!;+æ+æ ;[!;= = 96

14 b a - -a -b b a (a-b)(a+b-) - = -a -b (a-)(b-) a-b>0 a+b->0 a->0 b->0 (a-b)(a+b-) >0 (a-)(b-) b > a -a -b > <-++->->/ <-+->>;&; ;&;<< æ--+>> æ >;&; æ7 < æ <-+++3æ7 5æ7 æ-++3æ7 æ3 æ3 - æ3 3 a +b+c>0 <<3 a<0 a <<3 a(-)(-3)>0 a -a+3a>0 b=-a c=3a a -a+3a<5 D D =(-a) -a (3a-5)<0 a +5a<0 a(a+5)<0-5<a<0-8+7<0 [ -7+0æ0 <<7 æ5 < 5 <7 (-) - 0 [ æ æ (a +b )(c +d )-(ac+bd) =a d -abcd+b c =(ad-bc) æ0 (a +b )(c +d )æ(ac+bd) ad=bc 97

15 (+a)(a-)=0 a+0 <0 =-a =;a!; +3-=0 (+)(-)=0 =- = <0 =- æ0-3-=0 (+)(-)=0 =- = æ0 = =- = -(a-)+a-=0 yy -(a-)+=0 yy D D =(a-) -(a-)=0 (a-)(a--)=0 a= a=5 yy D D =(a-) -6=0 (a-+)(a--)=0 a=-3 a=5 yy a=5 3 a+b=- ab=3 a b 3 3[ -{ + }+ ]=0 a b ab a+b - + = = =;3!; a b ab 3 ab 3{ 3 ++=0 = = -=0 (-)( ++)=0 ++=0 + + =( ) +( ) +( ) =+ + =0 + = =- 5 cm y cm y cm " +y cm cm cm 6cm 98

16 +y+" +y = ;!;y=6 +y ={-(+y)} +y-y-7=0 y= +y=7 =3 y= = y=3 5 cm 6 a +b+c>0 -<<3 a<0 a -<<3 a(+)(-3)>0 a -a-3a>0 b=-a c=-3a b -a-c<0 -a -a+3a<0 -a +-3<0(+3)(-)<0 -;#;<< 7 f()= -(k+)+k f()= -(k+) +k=0 f() - 0 -(k+) k -k -k 0 (-)( +-k)=0 = +-k=0 +-k=0 +-k=0 = +-k=0 +-k=0 D D=+6k=0 k=-; 6; +-k=0 = +-k=0 k=;!; k -; 6;+;!;=; 6; 8 a + b + + =a+b+ + a b a b a+b+ + =a+;a!;+ +a ab= a b a ææ a ;a!;+æ a a =+= a + b + + a b a=b= 99

17 (+y){;[*;+;]!;} 5 3 {a+;a!;}{a+;a!;}æ9 f()æk f() k +y +y+3 +y +y+3 = +++y +y++ +y ++y--y =(+) +(y+) +(+y-) `æ0 0 +=0y+=0+y-=0 =y=0 >0y>0 8y 8y + ææ y y 8y = =y y y (+y){;[*;+;]!;} 5 +yæ'ƒ y ;[*;+;]!;ææ ;[*; ;]!; =y;[*;=;]!; y=;8{; 00

18 +k+k+=0 k ak+a p A={ p +p-36=0 } n(a) A={ + <a} B={ (-)(+)æ0} n(a;b)=3a 6 +(a+k)+k +a+k-=0 a k =0 a a+ a 7 a>b>0 'ƒa-b>'a-'b 0

19 +k +k+=k+=0 k=- k -=(-)=0 =0 = k+a=-+0=- ab ab=-36 ab (-36)(-36)(-8)(-8) (3-)(-3)(-9)(-9)(6-6) (+)( +-)=0 a +-=0 a a +a-=0 a+- =0 a {a+ }={a- }+=8 a a a+ a = ' y 8p p+yp=8p+y= 6p p +py =8p +y =8 = y= 5 A={ --a<<-+a} B={ - æ} 3 a=3 6 D D =(a+k) -(k +a+k-) =a +(k-)a-k+>0 a a +(k-)a-k+=0 D' D'=(k-) -(-k+) =k -3<0 '3 '3 - <k< k 7 ('ƒa-b) -('a-'b) =a-b-(a-'aåb+b) =('aåb-b) ='b('a-'b) a>b>0 'a>'b'b>0 'b('a-'b)>0 ('ƒa-b) >('a-'b) 'ƒa-b'a-'b 'ƒa-b>'a-'b ('ƒa-b ) >('a-'b) 'ƒa-b >'a-'b 50 % 50 % 0 % 0 % 0 % 0

20 -(k+)+k+3=0 k U A={5a -a-3}b={a-3a +a-} A;B={-}a a -a+=0 a (k+) +3 +(k+)-=0 k y=k [ +y = k < [ a< b - ++<0 ab 8 03

21 -(+)+k+3=0 D D =(k+) -(k+3)=k +k- =(k-)(k+)=0 k=- k= k - ab a+b=-ab=a -a+ =(a-b) =(a+b) -ab = -(a -a+) (a-) =0 a= 3 y=k- +y = -k+k -=0 D D=0 D =k -(k -)=-k +=0 k= -3-=(-)(+) =-(-)(+)<0 (-)(+)>0 <- > < a=b= ab=8 5 A;B={-} a -a-3=- a -a-=0 a= a=- a=b={-5} a=-b={--} A;B={-} a=- 6 (-)(+){ -(k+)+}=0 = k= =- k=-6 -(k+)+=0 k=3 k=-5 k - 7 <0--(-)<5-< >- -<<0 0 <-(-)<5<5 0 < æ+-<5<6 <3 <3 -<<3 8 b + a b b æ+æ ;ba; =8 a a 8 b ;ba;= a=b a 30 % <0 30 % 30 % 50 % ab 0 % 0 % 0

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